**Propagation of Acoustic Wave inside the Carbon **

**Nanotube: Comparative Study with Other Hexagonal **

**Material **

**Sanjay Srivastava **

Maulana Azad National Institute of Technology, Bhopal, India Email: [email protected]

Received September 28, 2012; revised October 30, 2012; accepted November 7, 2012

Copyright © 2013 Sanjay Srivastava. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**ABSTRACT **

Carbon nanotube is a novel and more explored material. In this paper, ultrasonic acoustic velocity of the carbon nano-tube has been calculated along unique axis at room temperature. For the evaluations of ultrasonic properties, second- and third-order elastic constants have been computed from Lennard-Jones interaction potential. Attenuation of ultra-sonic waves due to phonon-phonon interaction is predominant over thermoelastic loss. Carbon nanotube shows the unique behavior with the chiral number. Chiral number not only affect the band gap and tube radius of the carbon na-notube but also affect the mechanical properties like stiffness, bulk modulus, shear modulus of the tube. The peculiar behavior is obtained at 55˚ due to their least thermal relaxation time and highest Debye average velocity. Results are also compared with other hexagonal metallic materials which present in periods and group of the periodic table. They show the optimum behavior with other hexagonal materials.

**Keywords:** Stiffness; Young’s Modulus; Shears Modulus; Second- and Third-Order Elastic Constants; Acoustic
Velocity

**1. Introduction **

Carbon nanotube (CNT) are cylindrical carbon molecules
witch possess novel outstanding better properties (like
mechanical, electrical, thermal and chemical properties,
100 times stronger than steel, best field emission emitter,
can maintain current density more than 10−9_{A/cm}2_{, ther- }

mal conductivity comparable to that of the diamond)
which make then potential useful in a wide variety of the
applications (e.g., optical nano-electronics, composite
material, conducting polymers, sensor) [1,2]. Nanotubes
are composed of sp2_{ bonds, similar to those observed in }

graphite and they are naturally aligned themselves into ropes held together by Vander Waals force [3]. Then nono tubes are usually described using the chiral vector,

1 2

*h* , which connects two crystallographi-

cally equivalent sites (*A and* *A*¢) on a graphene sheet
(where a1 and a2 are unit vectors of the hexagonal hon-

eycomb lattice and *n *and *m *are integers). This chiral
vector * Ch* also defines a chiral angle

*õ*, which is the angle

between * Ch* and the zigzag direction of the graphene

sheet .Each nanotube topology is usually characterized
by these two integer numbers (*n*, *m*), thus defining some

*nxa* *mxa*

**C**

peculiar symmetries such as *armchair *(*n*, *n*) and *zigzag *

(*n*, 0) classes. The circumference vector or chiral vector
describes the structure of the CNTs. The structure of the
single-wall carbon nanotube (except for cap region on
both ends) is specified by a vector of original hexagonal
(also called hexagonal/honey Comb) lattice called the
chiral vector. The unusual high stiffness of CNT makes it
suitable ingredients for the application of several com-
posite materials. CNT shows the similar properties as
like other hexagonal materials (Ti, Y, Be, Mg, lantha-
nides ceramic etc.). Wave propagation velocity inside the
chosen materials is another key factor in ultrasonic char-
acterization, which in combination with attenuation can
provide important tools in understanding, the inspectabil-
ity of materials; for example, it can provide the informa-
tion about crystallographic texture, grain sizes, delecta-
bility of flaws, and microstructure. Ultrasonic velocity is
directly calculated from the elastic constants *i.e.* *V*

Elastic properties of materials at high pressure and high temperature are of great interest to researchers in many fields, such as physical sciences, earth sciences, and ma- terial sciences.

In the present investigation, we have chosen CNTs under the study due to their important features. In the present study, an applied approach for the calculation of second and third order elastic constants of the CNTs va- lidating the interaction potential model was calculated. These nonlinear elastic constants have been used to cal- culate the ultrasonic velocity and attenuation of the CNTs for their characterization. Their properties were compared with other hexagonal material and optimized results were produced with comparative study.

**2. Theory **

**2.1. Higher Order Elastic Constants**

A carbon atom in graphene will assemble in a single-
sheet hexagonal lattice resembling the surface of a hon-
eycomb. This is also known as a trigonal-planar *σ*-bond
framework, with an inter-atomic spacing, *acc*, of appro-
ximately 1.41Å along the bonds that are separated by 120
degrees. The 2p electrons from all the atoms on the sheet
constitute a cloud of delocalized π-orbital’s surrounding
the carbon cores, and these valence electrons, once ex-
cited, are responsible for conduction in graphene. Fur-
thermore, multiple sheets of graphene may assemble in
stacks, whereby two adjacent sheets are held together
weakly by dispersion forces and have an inter-layer
spacing of about 3.35 Å. The basis vectors

###

1*a* 3,0

**a**

and

###

###

2*a* 3 2,3 2

**a**

generate the graphene lattice, where *a* = 0.142 nm is the
carbon-carbon bond length. A and B are the two atoms in
the unit cell of graphene. The chiral vector begins and
ends at equivalent lattice points, so that the particular (*n*1;
*n*2) tube is formed by rolling up the vector so that its head

and tail join, forming a ring around the tube. The length
of *Ch *is thus the circumference of the tube, and the radius
is given by the formula

###

2 2###

1 2 1 2

3

2 2

*h* *cc*
*t*

*C* *a*

*R* *n* *n*

*n n* (1)

Many of the hexagonal crystals have axial ratio *p *= *c*/*a*
different from the ideal value 8

3

the lattice parameter in CNT ropes is

. A good estimate for

Lattice parameter 2 *Rt*3.4 inÅ (2)

Based on the knowledge of higher order elastic con- stants of the material is essential for the study of the an- harmonic properties of solids. Elastic constants also pro-

vide insight the nature of binding forces between the at- oms since they are represented by the derivatives of the internal energy. A complete set of elastic constants for materials (like second and third order elastic constant) is essential to estimate physical parameters such as Debye temperature, compressibility and acoustic anisotropy. The elastic energy density for a crystal of a cubic symmetry can be expanded up to quartic terms as shown below [6]:

2 3 4

1 2!

1 1

3! 4!

*ijk ij kl*

*ijklmn ij kl mn* *ijklmnpq ij kl mn pq*

*U U* *U* *U* *C x x*

*C* *x x x* *C* *x x x x*

(3)

where *Cijk *and *Cijklmn *are the second and third order elas-
tic constants in tensor form respectively. The second (*Cij*)
and third (*Cijk*) order elastic constants of material are de-
fined by following expressions.

2

, 1, ,

*ij*

*i* *j*
*U*

*C* *i j*

*e e*

6 (4)

3

, 1, ,

*ijk*

*i* *j* *k*
*U*

*C* *i j*

*e e e*

6 (5)

where, *U* is elastic energy density, *ei* = *eij* (*i* or *j* = *x*, *y*, *z*,
1, ,6

*i* _{} ) is component of strain tensor. Equations (1)
and (2) leads six second and ten third order elastic con-
stants (SOEC and TOEC) for the hexagonal closed pac-
ked structured materials [7]. In present approach, Len-
nard-Jones interaction potential

{

###

###

6###

7###

0 0

*r* *a r* *b r*

;

where *a*0, *b*0 are constants} was utilized due to many

body interaction potential. If it is assumed that the basic structure of carbon nanotube is a hexagon then Equation (1) leads following six second- and ten third-order elastic constants.

4

11 24.1

*C* *p C* 4

12 5.918

*C* *pC* 4

13 1.925

*C* *p C*

4

33 3.464

*C* *p C* 4

44 2.309

*C* *p* *C* 4

66 9.851

*C* *p C*

2 4

111 126.9 8.853

*C* *p B* *p C*

*p C*

*p C*

*B*

*B*

*B*

2 4

112 19.168 1.61

*C* *p B*

4 6

113 1.924 1.155

*C* *p B*

4 6

123 1.617 1.155

*C* *p B* *p C* 6

133 3.695

*C* *p* ,

6 155 1.539

*C* *p* *B* 4

144 2.309

*C* *p*

, *B*, 6

344 3.46

*C* *p*

2 4

222 101.039 9.007

*C* *p B* *p C* 6

333 5.196

*C* *p*

where

5

*C* *a p* , _{B}_{}_{}* _{a p}*3 3

_{, }

###

###

###

###

###

###

4###

1###

0

1 8 *n* 1 8 7

*o*

*nb n m* *a* *b a*

_{} _{} _{} _{}_{}

###

###

###

_{6}

*2*

_{a m n}_{6}

###

_{11}

_{a} _{4} 2

*m*,* n *= integer quantity; *b*0 = Lennard Jones parameter =

6.5 × 10–79_{ J}_{}_{m}7_{. }_{p = c/a}_{: axial ratio; }_{c }_{is the height of the }

elastic constants are related
rough Gruneisen parameters
unit cell and “*a*”be the basal plane distance. The symbol
“p” is the proposed parameter of axial ratio.

**2.2. Ultrasonic Velocity **

The second- and third-order to ultrasonic attenuation th

and acoustic coupling constant. On the basis of mode of lattice vibration, there are three types of velocities (lon- gitudinal, quasi shear and shear) in acoustical region [8]. The ultrasonic velocities can be computed using calcu- lated values of second-order elastic constants. These ve- locities depend on the direction of propagation of wave from the unique axis of hexagonal crystal [9]. The ultra- sonic velocities as a function of angle between direction of propagation and unique axis for hexagonal structured materials are given as follows [10]:

###

###

a parallellopiped by height c with a base which is a
rhombus of side “a”. A hexagonal crystal has one six-
fold axis of symmetry. All belong to the crystal class
P63/mmc. The ideal *c*/*a *ratio required for close packing
of spheres to form the HCP structure is 1.633 (*i.e.*

24 3). The tube radius is calculated from the Equation
(1). **Figure 1** shows the variation of tube radius with
chiral number. In curve *n*1 = varied but *n*2 is assumed to

be zero, *n*1 – *n*2 is the divisible of 3 (where *n*1 and *n*2 are

the chiral number of the CNT either in zigzag structure
or armchair structute). Hence, the tube is zigzag in nature
and they show metallic in character. In zigzag structure,
the chiral vector is oriented along the axis, hence in this
case the chiral angle *i.e.*, *ζ* = 0˚. In case of arm chair
structure, the chiral number is equal, but it is not zero.
The behaviour is again found in metallic nature, as
shown in **Figure 1(b)**. In **Figure 1(c)**, the difference is
not the divisible of 3; the found structure may be either
zigzag or armchair. However the behaviour of the CNT
is to be changed from metallic to semiconducting in na-
ture, while the bond angle *ζ* is 30˚. The variation of the
band bap with tube radius and chiral number in semi-
conducting CNT is shown in **Figure 2**. These band gaps
are calculated from the following formula:

2 2 2

2 44cos 66sin

*S*

*V* *C* *C* (7)

where *VL*, *VS*1and *VS*2 are longitudinal, qua

pure shear wave ultrasonic velocities. Variab

si shear and
les *ρ* and *θ*

2

*cc*
*g*

*t*
*t a*
*E*

*R*

represent the density of the material and angle with the
unique axis of the crystal respectively. The Debye aver-
age velocity (*VD*) is useful to study the Debye tempera-
ture and thermal relaxation time of the materials. The
Debye temperature (*TD*) is an important physical pa-
rameter of solids, which differentiate the classical and
quantum behavior of phonon [11]. The following expres-
sions have been used for the evaluation of Debye average
velocity and Debye temperature

###

###

The band gap of semiconducting CNT decreases with
increase the tube radius as well as chiral number. Know-
ledge of the carbon nanotube dispersion relation allows
one to identify whether a tube produces metallic or se-
miconducting behaviour, and in the latter case, determine
the conduction and valence band edges and thus the band
gap [12,13]. Typical values for the band gap are in the
range of tenths of an electron-volt to a few electron-volts,
and for the previously mentioned (16, 0) tube it is *~*0:62
eV. The parameters “*Rt*” for SWCNT (10, 10) is 6.7834
Å. Hence the lattice parameter “*a*” becomes equal to
16.9664 Å. By applying the symmetric operation, it is
also found that *C*11 = *C*22, *C*33, *C*44 = *C*55, *C*12, *C*13= *C*23,

and *C*66 can be obtained by measuring the velocity in

c-direction (six fold symmetry axis). The constants *C*11,
*C*33 and *C*44 have been determined from the direct relation,
*Cij* = *ρV*2_{, where }_{V}_{ is the relevant sound velocity. They }

are related with Equation (7). Seven velocity measure- ments enable us to determine all the seven second-order

1 3 2

6

*D*
*D*

*V* *n*

*T* *a*

*B*

*k* (8)
1 3

3 3 3

1 2

1 1 1 1

3

*D*

*L* *S* *S*

*V*

*V* *V* *V*

(9)

**3. Results and Discussion **

ure, conductivity and e, it is necessary to In order to examine the band struct

wave propagation inside the nanotub

derive its *ε*-*k* relation. The hexagonal unit cell is charac-
terized by *a* = *b* ≠ c, *α* = *β* = 90˚, *γ* = 120˚. The unit cell is

###

###

###

###

##

_{2}

##

1 22

2 2 2 2 2 2 2 2 2

33cos 11sin 44 11sin 33cos 44 cos sin 4cos sin 13 44 2

*L*

*V* _{}*C* *C* *C* _{}*C* *C* *C* _{} *C* *C* _{}

###

###

###

###

##

_{2}

##

1 22

2 2 2 2 2 2 2 2 2

1 33cos 11sin 44 11sin 33cos 44 cos sin 4cos sin 13 44 2

*S*

*V* *C* *C* *C* _{}*C* *C* *C* _{} *C* *C*

[image:3.595.121.286.495.567.2]**0** **5** **10** **15** **20** **25** **30**
**0.4**

**0.8**
**1.2**
**1.6**
**2.0**

**T**

**ube rad**

**ius (i**

**n **

**)**

**Chiral Number**
**(a) n**

**1=varied,n2=0**

**(b)n _{1}=n_{2}=varied**

**(c)n _{1}=6, n_{2}=varied=not equal to n_{1}**

**Figure 1. Variation of Tube radius with chiral number for **
**different values of ****n****1 and ****n****2.**

**0** **4** **8** **12** **16** **20** **24**

**0.2**
**0.3**
**0.4**
**0.5**
**0.6**
**0.7**
**0.8**
**0.9**
**1.0**
**1.1**

**Chiral Number**

**Tube R**

**a**

**di**

**u**

**s**

**0.1**
**0.2**
**0.3**
**0.4**
**0.5**
**0.6**
**0.7**
**0.9**

**(n**

**m**

**)**

**0.8**

**Ban**

**d**

** Ga**

**p**

**(eV)**

**Figure 2. Variation of Tube radius and Eg with chiral num- **
**ber (****n****1 = 6 ****n****2 = integers but not divisible by 3 and ζ). **
elastic constants. This is metallic in character and ultra

ed by sing lattice parameter and set of Equations (2) for (10, - sonic attenuation is due to phonon-phonon interaction. The SOECs and TOECs of SWCNT are calculat u

10) chiral system, their results are tabulated in **Table 1**.
Furthermore, the following physical parameters have
been also determined from set of equation. The other
parameter was calculated from following formula by the
evaluation of the *C*11, *C*12, *C*13, *C*33, *C*44, and *C*66:

###

11 12###

13 33The Bulk modulus*B*_{}2 *C* *C* 4*C* *C* _{} 9,

###

###

11 12 33 13 44 66

Shear modulus

2 4 12 3

*G*

*C* *C* *C* *C* *C* *C*

_{} _{} 0,

###

###

Young modulus *Y* 9*BG* 3*B G*

poission’s ratio

###

3*B*2

*G*

###

2 3*B G*

###

using s ond order elastic constants, which are tabulated in**Table 1**. The values of six independent elastic const s along with dependent variable were evaluated by calculating

on different deformation

All these elastic constants are positive and satisfy the
well-known Born’s criteria for mechanically stable hexa-
gonal crystals: *C* > 0, (*C*11 − *C*12) > 0, *C*44 > 0, and

ec

ant

the stress tensors s applied to the

equilibrium lattice of the hexagonal unit cell.

11

###

_{C}

_{C}###

_{C}_{2}

*2*

_{C}_{0}

11 12 33 12 . The variation of the elastic

ber.

The beautiful results a

in

constant with chiral number is shown in** Figure 3**. With
increase the chiral number, the elastic constant decrease
from higher to sudden lower values. Beyond (6, 10), the
values becomes constant with increase the chiral num
For zigzag structure, the maximum values are at higher
chiral number, but the optimum and useful values are
found from *n*1 = 12 to *n*1 = 21.

re tabulated in **Table 2** for
dif-ferent chiral number. The ultrasonic velocities are
evalu-ated using elastic constants and density which in turn
provides Debye average velocity in all chosen materials.
The variations of all the three types of velocity are shown
in **Figure 4**. With increase the angle from, the
longitudi-nal velocity increases from 0˚ to 60˚ but again decrease
with increase the angle of propagation. But shear velocity
direction-1 show the same effect in the opposite
direc-tion.

**3000**
**4000**
**5000**
**6000**
**7000**

**0** **5** **10** **15** **20** **25**

**0**
**1000**
**2000**

**C**

**(10**

**10N/**

**m**

**2)**

**11**

**Chiral Number**

**Figure 3. Variation of C11 with chiral number (****n****1 = 6 ****n****2 = **
**integers but not divisible by 3 and ζ). **

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**

**2000**
**3000**
**4000**
**5000**
**9000**

**6000**
**7000**
**8000**

**VL (m/s) **
**Vs1**
**Vs2**

**lo**

**c**

**ity**

** (m**

**/s**

**) **

**Angle (degree) **

**V**

**e**

**Table 1. SOECs (GPa) and TOECs (GP**

*C*11 *C*12 *C*13 *C*33 *C*44 66 *Y ν*

**a) of SWCNT at room temperature.**

*C* *B G *

42.43 10.42 9.34 46.31 11.20 16.64 21.04 14.74 35.85 0.34

*C*111 *C*112 *C*113 *C*123 *C*133 *C*344 *C*144 *C*155 *C*222 *C*333

−691.94 −109.71 −23.96 −30.45 −156.43 −146.65 −35.47 −23.64 −547.48 −60.62

**Table 2. Stiffness with chiral number. **

For Zigzag structure with metallic character

For arm chair structure with metallic character

For arm chair structure with Semiconducting behaviour

*n*1 *n*2 *C*11(N/m2) × 1010 *n*1 *n*2 *C*11(N/m2) ×1010 *n*1 *n*2 *C*11(N/m2) ×1010

6 0 6794.2 6 6 197.12 6 0 6794.2

9 0 531.93 9 9 9.6105 6 4 658.73

12 0 69.73 12 12 0.9473 6 7 110.55

15 0 12.969 15 15 0.1445 6 8 63.264

18 0 3. 22.054

21 0 0.86986 21 21 0.00 6 5.2699

24 0 28414 24 24 0.0 99409

27 0 0.010388 27 27 0.00 0.47064

64 30 0.0 1677

0667 18 18 0.0297 6 10

075 13

0. 0023 6 17 0.

0078 6 19

30 0 0.0041 3 30 000296 6 22 0. 1

The shear velocity in direction-2 cont

ses with erage Deby

for the at differen

un e a ow with ange i angle.

Th is th combined of the all e thre elocity wh h depends upon cond ord elasti nd the

de y o he materia e SWCNT ws similar

effect of t e velocity agation lik her t e other he ona materials. There are different types of the he ona aterials which are classi in a fferent gr and periods in t c table ey m n s, p d f ck eleme deal valu /a is 33. Be

and Mg = 1.5680 and M 6235 are the

s- k e ents, T d Hf e, O block

ember and from La to Tm are f-block elements. Only Tl

inuously increa- angle. The plot of av

propagation of wave

e *VD *(**Figure 5**)
t angles with
iqu xis sh large variation ch n

is e effect th e v

ic the se er c a

nsit f t ls. Th sho the

h l

prop e ot h

xag

xag l m fied di

oup he periodi . Th ay be i

d an blo nt. The i es c 1.6

(p 3 for Be g 1. 0)

bloc lem i, Y, Zr, an , R s,

d-m

is the p-block elements. For investigating the wave
propagation in the material, the density decides its nature.
The unit cell parameters “a” and “p” for Gd, Tb, Dy, Ho,
Er and Tm are 3.63Å, 3.60Å, 3.59Å, 3.58 Å, 3.56Å,
3.54Å and 1.592, 1.583, 1.574, 1.570, 1.570, 1.571
re-spectively. The value of *m*, *n* and b0 for these lanthanides

metals are 6, 7 and 2.3 × 10−64_{ erg cm}7_{ correspondingly. }

Three metals La, Pr and Nd have the DHCP (double HCP) structure where the stacking of close packed planes fol-lows the order ABACABAC instead of the usual HCP sequence ABAB [14]. Consequently, close packing un-der these conditions leads to an ideal c/a ratio of 3.3256. Overall, most of the metals were lower and within −0.1 of their ideal c/a ratio, except for Cd and Zn which ex-ceeded the ratio by 0.223 and 0.194, respectively. In case of transition metal like Ti, Y, Zr, and Hf, the SOEC shows the irregular behavior due involvement of 4f or-bital. In case of heavy rare earth elements, the SOEC

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**

**3900**
**4000**
**410**
**4600**

**0**
**4200**
**4300**
**4400**
**4500**

**Angle (degree) **

**Figure 5. Variation average Debye velocity vs. angle with **
**unique axis of crystal. **

are found to be increasing from metal Gd to Tm. Thus
the thermal softening behaviors are obtained as we move
from Tm to Gd. These metals have low bulk modulus
and higher order elastic constants in comparison to other
hexagonal structured material [7-10,12,13]. This reveals
that large strain will develop for a given deforming force
in these materials. **Figure 6** shows the variation of
aver-age Debye velocity vs. angle with unique axis of crystal
and compare with alkaline earth metal like Be and Mg
(because they possess HCP crystals structure). The
ve-locity of propagation in medium depends upon the elastic

**Av**

**er**

**a**

**g**

**e Ve**

**lo**

**ci**

**ty**

** (**

**m**

**/s**

**) **

constant, and density of the materials [15].

[image:5.595.284.538.181.534.2]**0** **10** **20** **30** **40** **50** **60** **70** **80** **9**
**4**

**5**
**6**
**7**
**8**
**9**
**10**
**11**

**0**

**A**

**v**

**er**

**ag**

**e Vel**

**o**

**ci**

**ty**

** (**

**m**

**/sx1**

**0**

**3** **)**

**Angle(degree)**
**carbon nanotube**
**Be**

**Figure 6. Variation average Debye velocity vs. angle with **
**unique axis of crystal and compare with alkaline earth **

**met-3**

oc

same crystals structure but they show dramatic effect on
the propagation of acoustic wave inside the materials. In
this material, the average Debye velocity decreases with
the angle and attain the minimum value around 45˚ - 55˚
and then increase with the angle, as shown in **Figure 8**.
Hf also possesses higher density (9 - 10 times heavy
from the carbon nanotube) as compared to carbon
nano-tube. Due to higher density, they show lower propagating
velocity inside the materials. Due to involvement of 4f
orbital in Hf and also expected double crystal structure
the propagating velocity along axis, they show negative
effect from the Ti, and Y. Re is again the member of the
5d series and exhibits the same crystal structure, but thei

city first decreases and attains the minimum values and

and very high elastic constant as compared to the known
**al (Be, ρ = 1.85 × 103 kg/m , and elastic constant ****C****11 = 292, **
**C****12 = 24, ****C****33 = 349, ****C****44 = 163, ****C****66 = 6 GPa). **

to five times less than “Be” but similar to Mg. it is evi-
dent from the **Figure 6** that the propagating velocity in-
side the Carbon nanotude is two times less than the Be
crystal structure. Ti, Co, and Zn belong in 3d series and
Y and Zr in 4d series but they possess HCP crystal
structure. The variation of the average velocity with
propagating angle along the unique axis is presented in
the **Figure 7**. The density as well as second order elastic
constant of the Ti and Y is high as carbon nanotube but
the velocity of propagation along the crystals axis shows
negative effect. To compare with carbon nanotube the
average velocity in Ti decrease with increasing angle, as
evident from **Figure 7(a)**. Y show similar effect as Ti,
compared with the **Figure 7(b)**. Hf is the member of
d-bl k (5d) element like Ti, Y and also possesses the

r
trends for propagating velocity inside the materials show
positive effect from the Hf , Ti and Zr. **Figure 9** shows
the variation of average Debye velocity with the
propa-gating axis. It is evident from the **Figure 9** that the
ve-lo

then increase with increase the angle. Due to their high density (19 - 20 times heavy from the carbon nanotube)

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**
**3.8**

**3.9**
**4.0**
**4.1**
**4.2**
**4.3**
**4.4**
**4.5**
**4.6**

**A**

**v**

**er**

**age**

** V**

**e**

**lo**

**ci**

**ty**

**(m**

**/s**

**x10**

**3** **)**

**Angle(degree)**
**Carbon Nanotube **
-Ti

(a)

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**
**2.0**

**2.5**
**3.0**
**3.5**
**4.0**
**4.5**

**A**

**v**

**er**

**age v**

**el**

**o**

**ci**

**ty**

** (**

**m**

**/sx10**

**3)**

**Angle (degree)**

**Carbon nanotube **
-Y

(b)

**Figure 7. Variation average Debye velocity vs. angle with **
**unique axis of crystal and compare with transition metal (a: **
**Ti, ρ = 4.51 × 103 kg/m3**

**, and elastic constant ****C****11 = 160, ****C****12 **
**= 90, ****C****33 = 181, ****C****44 = 46.5, ****C****66 = 66 and b: ****Y****, ρ = 4.47 × 103 **

**3**

**kg/m , and elastic constant ****C****11 = 77.9,**** C****12 = 29.2, ****C****33 = 76.9, **
**C****44 = 24.3, ****C****66 = 20 GPa). **

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**
**2.62**

**2.64**
**2.66**
**2.68**
**2.70**
**2.72**
**2.74**
**2.76**

Hf

**A**

**v**

**e**

**rage V**

**e**

**lo**

**c**

**it**

**y**

** (**

**m**

**/s**

**x1**

**0**

**3** **)**

**Angle(degree) **

**Figure 8. Variation average Debye velocity vs. angle with **
**unique axis of crystal and compare with transition metal **
**(Hf, ρ = 13.1 × 103 kg/m3**

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**
**3.68**

**3.70**
**3.72**
**3.74**
**3.76**
**3.78**
**3.80**
**3.82**

**Re**

**A**

**v**

**erag**

**e V**

**e**

**locity**

**(m**

**/s)**

**Angle(degree) **

**Figure 9. Variation average Debye velocity vs. angle with **
**unique axis of crystal and ** **pare with transition metal **

In periodic table and known HCP crystal structure, Os
is the heaviest element among all the elements. **Figure **
**10** shows the variation of average Debye velocity with
propagating angle along the crystal axis. It can be seen
from the **Figure 10** that the velocity decrease with
in-crease the angle, beyond attending the minimum velocity
they again increase. Out of the various materials studied
in this research work, they possess the higher values
be-cause of the higher elastic constant along with the
maxi-mum density (18 - 20 times more than the carbon nano-
tube).

The group Os, Ru, Re, Be and Co exhibit the highest
moduli of all the HCP metals. All show a pronounced
maximum value of *E *on the basal plane, for which *θ* is
zero, and a tendency by Ru, Re and Co to exhibit a min

cating *G *is
high-est on the basal plane. The elastic constants are
impor-tant since they are related to hardness and are used for
the determination of the ultrasonic velocity. The plot of

*VD *(**Figure 11(a)**) for the propagation of wave at
differ-ent angles with unique axis implies that these metals
have slight variation in velocity for each direction of
propagation and have maxima at *θ*= 55˚. Fig.11b shows
the variation of the average velocity with crystals axis or
the rare earth elements along with carbon nanotube.
Lanthanides possess higher velocity with carbon
nano-tube because of similar elastic constant and heavier
den-sity. Carbon nanotube is the allotrope of the carbon and

**com**
**(Re, ρ = 21 × 103 kg/m3**

**, and elastic constant ****C****11 = 616, ****C****12 = **
**273, ****C****33 = 683, ****C****44 = 161, ****C****66 = 206 GPa). **

metal in HCP crystals structure expect Os, they exhibit lower velocity because of density effect, evaluated from Equation (7).

-imum between 0 < *θ* < 90. With the exception of Be, the

*G*-behaviour exhibits a minimum when *θ* is zero and 90˚,
and a maximum for 0o < *θ* < 90˚. In the case of Be, a
maximum *G *occurs when *θ* is zero, indi

Tl belongs to the same family. The pπ-pπ interacti n
provides the propagation of the velocity inside the metal.
In other case, s*σ*-s*σ*, dπ-sπ dπ-dπ, dπ-fπ and fπ-fπ are

**4.56**
**4.58**

o

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**

**4.38**
**4.40**
**4.42**
**4.44**
**4.46**
**4.48**
**4.50**
**4.52**
**4.54**

**Os**

**A**

**v**

**erag**

**e Vel**

**oc**

**Angle(degree)**

**Figure 10. Variation average Debye velocity vs. angle with **
**unique axis of crystal and compare with transition metal **
**(Re, ρ = 22.61 × 103 g/m3**

**, and elastic constant ****C****11 = 765, ****C****12**
**= 229, ****C**

**it**

**y**

**(m**

**/s**

**x10**

**3** **)**

** = 846, ****C**** = 270, ****C**** = 219 GPa). **

**33** **44** **66**

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**
**4.6**

**4.8**
**5.0**
**5.2**
**5.4**
**5.6**
**5.8**
**6.0**
**6.2**
**6.4**
**6.6**

**A**

**v**

**er**

**a**

**g**

**e**

** V**

**elocit**

**y**

**(m**

**/s **

**x1**

**0**

**3** **)**

**Angle (degree)**

**G d**
**Tb**
** D y**
**H o**
**E r**
**Tm**

(a)

**4 .5**
**5 .0**
**5 .5**
**6 .0**
**6 .5**

**0** **1 0** **2 0** **3 0** **4 0** **5 0** **6 0** **7 0** **8 0** **9 0**

**4 .0**

**A n g le (d e g re e )**
**c a rb o n n a n o tu b e**
**G d**

**T m**

**A**

**v**

**erage**

** Velo**

**cit**

**y**

** (**

**m**

**/sx1**

**0**

**3** **)**

(b)

**Figure 11. Variation average Debye velocity vs. angle with **
**unique axis of crystal of (a) Heavy rare earth elements and **
**(b) compare with carbon nanotube. **

planar hexagon structure. Another predominant effect which lowers the propagation of the acoustic wave inside

he carbon naotube is the localization of t t

tr

he bound
elec-on in the belec-onding state. Its propagatielec-on can be
com-pared with Tl, because of the same member. They are
much heavier and possess same stiff. Their variation of
velocity with angle is shown in **Figure 11**. It is evident
from the **Figure 12** that they show two humps at the
dif-ferent angle, and also show the lower velocity with cabon
nanotude.

**4. Conclusion **

On the basis of the above study, the following conclu- sions have been drawn:

1) Carbon nanotube are characterized by the chi number and tube radius inc ut band gap of the car- bon tube decrease decreases with increasing the chiral number.

2) Lennard-Jones interaction potential was used to calculate the second and third stiffness of the carbon na-notube.

3) A detailed study of the elastic properties of carbon nanotube gives complete set of second order elastic stiff-ness constants, compliance constants, Poisson’s ratios, volume compressibility and bulk modulus of this crystal. All these important parameters for this crystal are meas-ured and tabulated. The elastic anisotropy of various pa-rameters like phase velocity, slowness, Young’s modul s and linear compressibility are depicted in the form of two

bon nanotube and de-ral rease b

u

dimensional surface plots.

4) The mechanical property of the carbon nanotube depends upon chirality of the car

creases with increasing the number. Also it is found that

*C*33 is less than *C*11, which corresponds to the binding

forces along the basal plane of the crystal. Second and

**0** **10** **20** **30** **40** **50** **60** **70** **80** **90**

**1000**
**1200**
**1400**
**1600**
**1800**
**2000**
**2200**

**Angle (degree)**

**n average Debye velocity vs. angle with **
**compare with p-block-metal (Re, **

**33** **44** **66**

t are utilized to determine the D

oms along the c-direction. Not only this pa-ra

s calculated with the help from second order el

[1] A. Javey, H. Kim, M. Brink, Q. Wang, A. Ural, J. Guo, P.
Mcientyre, P. McEuen, M.Lundstrom and H. Dai, “High
K-Dielectric for Advanced Carbon Nanotube Transistors
and Logic Gates,” *Nature Materials*,Vol. 1, No. 1, 2002,
pp. 241-246.

[2] J. Wildoer, L. Venema, A. Rinzler, R. Smalley and C.
Dekker, “Electronic Structure of Atomically Resolved
Carbon Nanotubes,” *Nature*, Vol. 391, No. 6662, 1998,
pp. 59-62. doi:10.1038/34139

**Aver**

**age Vel**

**o**

**ci**

**ty(**

**m**

**/s)**

**Figure 12. Variatio**
**unique axis of crystal and **
**ρ = 11.85 × 103 kg/m****3**

**, and elastic constant ****C****11 = 765, ****C****12 = **
**229, ****C**** = 846, ****C**** = 270, ****C**** = 219 GPa). **

third order elastic constan ebye average velocity.

5) Therefore, in carbon nanotube, the bonding between the atoms along the a-b plane is stronger than that be-tween the at

meter, but also second and third order parameter gives the same indication. The anisotropic properties of SWCNT

astic constant are well related with the ultrasonic ve-locities as well as with the higher order elastic constants.

6) The highest change in Debye average was observed at 50˚ - 55˚along the unique axis.

7) Due to change in bonding from metal to SWCNT, the observed changes in velocity with different type of the metal along with CNT are different at the different angle. Their values depend upon the phonon-phonon vibration in the lattice.

8) In metallic material, phonon-phonon vibration is easily achieved due acoustic wave interaction as com-pared to CNT because of less tightly packing.

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